I hate this too.
I think the best thing in future it to be insistent and maybe mention to the other parent quietly it's something you are working on with DD - that she can sometimes struggle with the idea of only playing with toys and not taking them home (whether true or not!) and that you're trying to teach her to become more comfortable with leaving a place and the toys there behind. That way you're not seeming ungrateful but explaining it's a teaching moment that's necessary.
You can say it's really kind but it's better for DD to get used to the fact she can't take things home that aren't hers. That it's ok that she's upset but we aren't taking anything back and we have plenty of toys at home. Then carry/hurry her out crying if needed, before there's time for debate, while remaining upbeat, polite and grateful for a nice time.
It is really annoying that now you have to remember the toy, keep an eye on it, make sure it's looked after and now arrange a play date you didn't actually want. I generally find lending things uncomfortable anyway.
Also it really is true that we do not l let children be sad often enough.
It's understandable a child is sad about leaving when they were having a nice time with their friends and toys. It's not a bizarre reaction at all. As we get older we learn to manage the feelings and they're not as intense but it's a perfectly normal emotion to feel. I feel a little sad when I leave a place I'm enjoying.
Emotions are useful. As she grows up, that feeling teaches her what she likes and values and encourages her to take actions to maintain friendships that give her joy so she can continue spending time with people. The feeling is unpleasant at the time but even unpleasant emotions are useful and they pass. To always jump to fixing it and making them happy again, like this other parent did, is to teach children that negative emotions are to be feared and avoided. Sometimes they don't need to be fixed. It's ok that they're just there for a while.
I say this somewhat hypocritically as it's also my automatic instinct to do the same sort of fixing with my own child and it's very challenging to notice and stop myself.
I think your best bet is swift goodbyes once the tears happen and help DD through it once you're alone.